Question

The direction cosines of the normal to the plane containing the lines having direction ratios $1,2,1$ and $4,5,-3$ are

• A. $\frac{-11}{\sqrt{179}},\frac{7}{\sqrt{179}},\frac{-3}{\sqrt{179}}$
• B. $\frac{1}{\sqrt{2}},0,\frac{-1}{\sqrt{2}}$
• C. $\frac{5}{\sqrt{41}},\frac{-4}{\sqrt{41}},0$
• D. $\frac{2}{\sqrt{5}},\frac{-1}{\sqrt{5}},0$

Answer 1

Answer: (A)

Let $b_1=\hat{i}+2\hat{j}+\hat{k}$ and $b_2=4\hat{i}+5\hat{j}-3\hat{k}$
Normal vector to plane
$b_1\times b_2=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& 1\\ 4& 5& -3\end{array}\right|$
$=\hat{i}(-6-5)-\hat{j}(-3-4)+\hat{k}(5-8)$
$=-11\hat{i}+7\hat{j}-3\hat{k}$
$\therefore$ Direction cosines of normal to the plane
$=\frac{-11}{\sqrt{(-11{)}^2+7^2+(-3{)}^2}},\frac{7}{\sqrt{(-11{)}^2+7^2+(-3{)}^2}}$
$\frac{-3}{\sqrt{(-11{)}^2+7^2+(-3{)}^2}}=\frac{-11}{\sqrt{179}},\frac{7}{\sqrt{179}},\frac{-3}{\sqrt{179}}$